Finite Group Databasebeta

← #13099915

Subgroups — #13099915

Order 1600.

248 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717451C211yesmaximal, characteristic
2C2×#87856C211yesmaximal, characteristic
3#11716936C211yesmaximal, characteristic
4C2×C4×C5⋊D5C2211yescharacteristic
5C22⋊#1044C412yes—
6C4⋊#1044—21no—
7C4×#1044—21no—
8C2×#6033C412yescomplemented
9#87856—21no—
10#87856—21no—
11C5×#681C412yescomplemented
12#87769—21no—
13#87767—21no—
14#64902—52nomaximal
15C22×C5⋊D5C2×C411yescharacteristic
16C2×#1044—22no—
17C2×C4×C52C2×C411yescharacteristic
18C2×#1039C2×C411yescharacteristic
19C4×C5⋊D5D411yescharacteristic
20C4×C5⋊D5D411yescharacteristic
21C4×C5⋊D5D411yescharacteristic
22C4×C5⋊D5Q811yescharacteristic
23#6033—22no—
24C5×#168—22no—
25#4130—52no—
26C2×#700—52no—
27C2×#700—102no—
28D5×M16—52no—
29C22×C52C4211yescharacteristic
30C2×C5⋊D5—41no—
31C2×C5⋊D5C22⋊C411yescharacteristic
32C2×C5⋊D5C4⋊C411yescharacteristic
33#1044—42no—
34C4×C52C22⋊C411yescharacteristic
35C4×C52C4⋊C411yescharacteristic
36#1039C22⋊C411yescharacteristic
37#1039C4⋊C411yescharacteristic
38C2×C4×D5—52no—
39C2×C4×D5—102no—
40C22⋊AGL(1,𝔽5)—54no—
41C2×#170—54no—
42C2×#170—104no—
43C4⋊AGL(1,𝔽5)—102no—
44C4×AGL(1,𝔽5)—102no—
45#700—102no—
46#700—102no—
47#700—104no—
48#700—104no—
49C5×M16—52no—
50#681AGL(1,𝔽5)12yescomplemented
51#676—102no—
52C8×D5—102no—
53#344—251noSylow
54C2×C52—21no—
55C2×C52#9511yescharacteristic
56C5⋊D5—21no—
57C5⋊D5—21no—
58C22×D5—52no—
59C22×D5—102no—
60C2×AGL(1,𝔽5)—104no—
61C2×C4×C5—22no—
62C2×C4×C5C2×AGL(1,𝔽5)12yes—
63C2×Dic5—52no—
64C2×Dic5—102no—
65C4×D5—52no—
66C4×D5—52no—
67C4×D5—52no—
68C4×D5—52no—
69C4×D5—102no—
70C4×D5—102no—
71C4×D5—102no—
72C4×D5—102no—
73#170—104no—
74#170—108no—
75C5×C8—102no—
76#168—22no—
77C2×M16—251no—
78C2×M16—251no—
79#117—251no—
80C52#34411yescharacteristic, complemented, Sylow
81C22×C5—22no—
82C22×C5C4×AGL(1,𝔽5)12yes—
83C2×D5—52no—
84C2×D5—52no—
85C2×D5—102no—
86C2×D5—102no—
87C2×D5—202no—
88C2×D5—204no—
89AGL(1,𝔽5)—204no—
90C4×C5—22no—
91C4×C5—22no—
92C4×C5C4⋊AGL(1,𝔽5)12yes—
93C4×C5C22⋊AGL(1,𝔽5)12yes—
94Dic5—52no—
95Dic5—52no—
96Dic5—102no—
97Dic5—102no—
98C22×C4—251no—
99M16—52no—
100M16—501no—
101M16—501no—
102M16—501no—
103C2×C8—252no—
104C2×C8—501no—
105C4⋊C4—501no—
106C22⋊C4—252no—
107C42—501no—
108C2×C5—24no—
109C2×C5—22no—
110C2×C5—22no—
111C2×C5(C2×AGL(1,𝔽5))⋊C412yes—
112D5—102no—
113D5—102no—
114D5—202no—
115D5—202no—
116C23—251no—
117C2×C4—251no—
118C2×C4—251no—
119C2×C4—251no—
120C2×C4—251no—
121C2×C4—251no—
122C2×C4—502no—
123C2×C4D5⋊AGL(1,𝔽5)11yescharacteristic
124C8—102no—
125C8—502no—
126C5—22no—
127C5#6490212yescomplemented
128C22—251no—
129C22—251no—
130C22—1001no—
131C22#8783011yescharacteristic
132C4—251no—
133C4—251no—
134C4—1002no—
135C4#8783111yescharacteristic
136C4#8783211yescharacteristic
137C2—21no—
138C2—501no—
139C2—501no—
140C2#1171677611yescharacteristic
subgroupG/H
centerC2#11716776
commutatorC22×C52C42
Frattini, hypercenterC2×C4D5⋊AGL(1,𝔽5)
FittingC2×C4×C52C2×C4
socleC2×C52#95
prime radicalC22×C5⋊D5C2×C4
cosocleC2×C4×C5⋊D5C22
hypocenterC52#344
LCSG ▷ C22×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
8. C2×#6033C4semidirect
11. C5×#681C4semidirect
50. #681AGL(1,𝔽5)semidirect
80. C52#344semidirect
127. C5#64902semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117174512
C2×#878562
#117169362
#649025